1 Answer
1

$$\frac{\left(p_1+p_4\right)\left(p_2+p_3\right)}{\left(p_1+p_3\right)\left(p_2+p_4\right)} = 1 + \frac{\left(p_2-p_1\right)}{\left(p_1+p_3\right)} \times \frac{\left(p_4-p_3\right)}{\left(p_2+p_4\right)} $$ and the right hand side is greater than or equal to $1 + 0 \times 0$ but less than $1 + 1 \times 1$.

The values $(1,n,n,n^2)$ achieve the lower bound when $n=1$ but approach the upper bound when $n$ increases without limit: $n=400$ gives a figure over $1.99$.

Just for completeness: Thus, the answer to the first question is that the expression has no maximum.
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jorikiMar 13 '12 at 1:23

@joriki,@Henry: You mean, it has no maximum but it is upper bounded by 2, right?
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HeliumMar 13 '12 at 14:32

@Mohsen: Yes. In fact its supremum is $2$. By the way, you don't need to ping Henry when you comment under his answer; the author of a post is automatically notified of any comments under it.
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jorikiMar 13 '12 at 14:45